Polybutylene terephthalate-based resins are excellent in chemical resistance, heat resistance, and mechanical properties and are widely used as industrial resins. However, they are crystalline and are thereby poor in flexibility and paintability, resulting in limitation of the purpose of use. Accordingly, it is proposed a method (Patent Document 1) for imparting flexibility to a polybutylene terephthalate-based resin by adding a small amount (less than 10% by mass) of a thermoplastic elastomer, while maintaining the original characteristics of the polybutylene terephthalate-based resin, but the resulting flexibility is insufficient. In addition, it is proposed a composition (Patent Document 2) having enhanced rubber elasticity and flexibility by blending a rubber softener and a hydrogenated derivative of a block copolymer consisting of a special monovinyl-substituted aromatic hydrocarbon and a conjugated diene to a polyester elastomer composed of a block copolymer consisting of an aromatic polyester and a polyether, but the enhancement of both the paintability and the oil resistance is still insufficient. Thus, a polymer or a polymer composition that has both sufficient paintability and sufficient oil resistance and also can provide a molded article excellent in flexibility has not been currently obtained.    [Patent Document 1] Japanese Unexamined Patent laid-open No. 2006-225413    [Patent Document 2] Japanese Patent No. 2817879